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ALGEBRA II
LESSON 5-6
RADICAL EXPRESSIONS
Pg. 250-255
Objectives:
1. Be able to simplify radical expressions
2. Add, subtract, multiply or divide radical expressions accurately
3. Find solutions to problems using radical expressions and future equations
STEPS TO SIMPLIFY RADICAL EXPRESSIONS:
3 ______
√ 108
1) Identify factors of the radicand (number under the radical sign) using basic numbers raised to
the power of the index (Factors of 108 based on cubed power are 27 (33) times 4)
2) Isolate the perfect factor, then simplify and recombine results
3 ____
3
____
√ 108 = (108)1/3 = (27)1/3 times 4(1/3) = 3 times 4(1/3) = 3√ 108
NOTE: When dealing with even numbered index (the small number above) be sure both factors are
non-negative! Why? Because the even root of any negative number is an imaginary number.
Example 1: Simplify 2 _______
______________
__
√25a b = √5 (a ) (b ) b = 5
4 9
2
2 2
4 2
2/2
2 2/2
4 2/2
1/2
(a ) (b ) b
= 5a b √ b
2
4
THE DIVISION OF RADICAL PROCESS
NOTE: Answers must NOT have radicals in the denominator! Therefore, we must rationalize the
denominator to eliminate the radical. (In real words, multiply the denominator by another radical (in the
form of 1/1 to form a perfect square, cube, etc depending on the index to eliminate the radical on the
bottom!
Remember this? (1/2)(2/2) = 2/4 We didn’t change the value of the fraction, just the looks. We will
use this same principle to eliminate the radical from the denominator of a fraction.
____
___
__
__
___
___
Example 2 √ 7
= √ 7 = √7 times √5 = √35 = √35
5
√5
√5
√5 √25
5
NOTE: Any radical of index “2” called the square root, when multiplied by itself will yield the radicand
(the number under the radical.) or in our example the “5”
** Remember to factor the numerator if at all possible.
What if the index is a number other than 2? Then we need to determine the value of the radicand to
the power of the index. Usually multiply the top and bottom by the radicand to a power that make the
resulting denominator power divisible by the index under the radical. This will yield a value of the
radicand that will “magically” factor out of the radical.
This will require the assistance of an example to illustrate the importance of the changing index.
5____
y8
Example 3 √
=
x7
5___
y8
√
√ x7
3__
Try this problem: √2
9x
=
__
_____
5__
5 ___
8/5 3/5
8
3
8
3
= √ y times √ x = √ y x = y x = y √y3x3
√ x7
√ x3 √ x10
x10/5
x2
5
3__
2
√6x
3x
MULTIPLYING RADICALS
3_____
Think of this as a single number: 5 √100a2
(as one unit together) then multiplied by another single
unit (both having the same index)
3 ____
3 ___
3 ______
3 ______
5√100a2 times √10a =
5√1000a3 =
5√103 a3 = 5(10)(a) = 50a
ADDING OR SUBTRACTING RADICALS
Note: To either add or subtract radicals, BOTH INDEX and RADICALS must be alike in both terms
to combine the results into one unit.
__
__
__
__
__
___
2 √3 + 3 √3 = 5√3
OR 6√17 - 4√17 = 2√17
MULTIPLYING SUMS OR DIFFERENCES OF RADICALS
(FOIL and combine like terms as necessary)
__
_
__
(2√3 + 3√5)(3 – √3 ) =
F
O
I
L
(2√3)(3) + (2√3)(- √3) + (3√5)(3) + (3√5)(- √3)
6√3 +
- 2√9
+ 9√5
+ - 3√15
6√3 + (-2)(3) + 9√5 + (-3)√15
__
__ __
= 6√3 – 6 + 9√5 - 3√15
Rationalize the Denominators using a Conjugate of the Denominator
Conjugate: A Binomial of the form (a + b)(a – b) if when “FOIL”ed will eliminate the center term
This process, when used with radicals, effectively eliminates the radical by squaring the first and last
terms, while the middle two terms cancel each other out (like the Difference of Two Squares)
Example 4
2 + √3 times 4 + √3
4 - √3
4 + √3
= 8 + 2√3 + 4√3 +(√3)2 = 8 + 3 + 6√3 = 11 + 6√3
16 + 4√3 – 4√3 - (√3)2
16 – 3
13
NOTE: 11 + 6√3 ≠ 17√3  It’s like adding 11 + 6x with only “like terms” combining!